Computation of Jacobi sums of order l^2 and 2l^2 with prime l

TL;DR

This article presents fast computational algorithms for Jacobi sums of orders l^2 and 2l^2 with odd prime l and implements two additional algorithms which demonstrate the minimality of cyclotomic numbers required for the determination of all Jacobi summed orders.

Abstract

In this paper, we present the fast computational algorithms for the Jacobi sums of orders $l^2$ and $2l^{2}$ with odd prime $l$ by formulating them in terms of the minimum number of cyclotomic numbers of the corresponding orders. We also implement two additional algorithms to validate these formulae, which are also useful for the demonstration of the minimality of cyclotomic numbers required.

Theorems & Definitions (12)

• Theorem 4.1
• proof
• Theorem 4.2
• proof
• Remark 4.1
• Remark 4.2
• Remark 4.3
• Theorem 4.3
• proof
• Remark 4.4